The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $22.3$ years; the standard deviation is $5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living longer than $17.3$ years.
Solution: $22.3$ $17.3$ $27.3$ $12.3$ $32.3$ $7.3$ $37.3$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $22.3$ years. We know the standard deviation is $5$ years, so one standard deviation below the mean is $17.3$ years and one standard deviation above the mean is $27.3$ years. Two standard deviations below the mean is $12.3$ years and two standard deviations above the mean is $32.3$ years. Three standard deviations below the mean is $7.3$ years and three standard deviations above the mean is $37.3$ years. We are interested in the probability of a snake living longer than $17.3$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the snakes will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the snakes will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $17.3$ years and the other half $({16\%})$ will live longer than $27.3$ years. The probability of a particular snake living longer than $17.3$ years is ${68\%} + {16\%}$, or $84\%$.